Nonlinear model reduction of dynamical power grid models using quadratization and balanced truncation

Tobias K. S. Ritschel; Frances Weiß; Manuel Baumann; Sara Grundel

doi:10.1515/auto-2020-0070

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Nonlinear model reduction of dynamical power grid models using quadratization and balanced truncation

Tobias K. S. Ritschel
Tobias K. S. Ritschel received a B. Sc. degree in mathematics and technology in 2013 and a M. Sc. degree in mathematical modelling and computation in 2015 from Technical University of Denmark. In 2018, he received a Ph. D. degree in applied mathematics, also from Technical University of Denmark. Since 2020, he has been holding a position as postdoctoral researcher at the Max Planck Institute for Dynamics of Complex Technical Systems, Germany.
, Frances Weiß
Frances Weiß received a Diploma in political science from Free University Berlin and University Oslo in 2009. In 2015 and 2019, she received a B. Sc. and M.Sc in science mathematics, respectively, from Otto von Guericke University Magdeburg. Since 2020, she has been holding a position as research assistant at Fraunhofer Institute for Transportation and Infrastructure Systems, Germany.
, Manuel Baumann
Manuel Baumann received his Ph. D. in applied mathematics from Delft University of Technology in 2018. Prior to that, he studied mathematics at Technical University of Berlin (B. Sc.) and computer science at Royal Institute of Technology in Stockholm and at Delft University of Technology (M. Sc., double degree program). His research interests include numerical linear algebra, computational geophysics, and model-order reduction and optimization of power systems. He has published several journal papers in these fields including in SIAM Journal on Scientific Computing and Springer Computational Geosciences. From 2018 to 2019, he was a postdoctoral researcher at the Max Planck Institute for Dynamics of Complex Technical Systems, and since 2020, he has been holding a position as research scientist at Philips GmbH Innovative Technologies, Germany.
and Sara Grundel
Sara Grundel was born in Munich Germany in 1981. She received a Diploma in mathematics from ETH Zurich in 2005 and a Ph. D. degree in mathematics from the Courant Institute of Mathematical Sciences at New York University in New York City in 2011. Since then she has been holding a postdoctoral position at the Max Planck Institute in Magdeburg, Germany. In 2015 she became a research team leader focused on the simulation of energy system.

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From the journal at - Automatisierungstechnik Volume 68 Issue 12

Abstract

In this work, we present a nonlinear model reduction approach for reducing two commonly used nonlinear dynamical models of power grids: the effective network (EN) model and the synchronous motor (SM) model. Such models are essential in real-time security assessments of power grids. However, as power grids are often large-scale, it is necessary to reduce the models in order to utilize them in real-time. We reformulate the nonlinear power grid models as quadratic systems and reduce them using balanced truncation based on approximations of the reachability and observability Gramians. Finally, we present examples involving numerical simulation of reduced EN and SM models of the IEEE 57 bus and IEEE 118 bus systems.

Zusammenfassung

Dieser Artikel beschreibt einen Ansatz zur nichtlinearen Modellordnungsreduktion zweier häufig benutzter Modelle zur Beschreibung dynamischer Stromnetze, das Effektive Netzwerk (EN) Model und das Synchroner Motor (SM) Model. Solche Modelle sind essentiell in der Sicherheitsanalyse von Stromnetzen. Zur Echtzeitanalyse benötigt man für große Netze allerdings Reduktionsmethoden. Wir schreiben das nichtlineare System in ein quadratisches um, welches dann mit Hilfe von balanciertem Abschneiden, basierend auf der Steuerbarkeits- und Beobachtbarkeitsgramschen, durchgeführt wird. Im Anschluss werden numerische Simulationen der reduzierten EN und SM Modelle am IEEE 57 und IEEE 118 Bus Beispiel gezeigt.

Keywords: nonlinear model reduction; balanced truncation; dynamical power grid models; quadratic systems

Schlagwörter: nichtlineare Modellreduktion, balanciertes Abschneiden, dynamische Stromnetzmodelle, quadratische Systeme

Funding source: Bundesministerium für Bildung und Forschung

Award Identifier / Grant number: 05M18EVA

Funding statement: We acknowledge the financial support from the German Federal Ministry of Education and Research in the project KONSENS: Konsistente Optimierung und Stabilisierung Elektrischer Netzwerksysteme (BMBF grant 05M18EVA).

About the authors

Tobias K. S. Ritschel

Tobias K. S. Ritschel received a B. Sc. degree in mathematics and technology in 2013 and a M. Sc. degree in mathematical modelling and computation in 2015 from Technical University of Denmark. In 2018, he received a Ph. D. degree in applied mathematics, also from Technical University of Denmark. Since 2020, he has been holding a position as postdoctoral researcher at the Max Planck Institute for Dynamics of Complex Technical Systems, Germany.

Frances Weiß

Frances Weiß received a Diploma in political science from Free University Berlin and University Oslo in 2009. In 2015 and 2019, she received a B. Sc. and M.Sc in science mathematics, respectively, from Otto von Guericke University Magdeburg. Since 2020, she has been holding a position as research assistant at Fraunhofer Institute for Transportation and Infrastructure Systems, Germany.

Manuel Baumann

Manuel Baumann received his Ph. D. in applied mathematics from Delft University of Technology in 2018. Prior to that, he studied mathematics at Technical University of Berlin (B. Sc.) and computer science at Royal Institute of Technology in Stockholm and at Delft University of Technology (M. Sc., double degree program). His research interests include numerical linear algebra, computational geophysics, and model-order reduction and optimization of power systems. He has published several journal papers in these fields including in SIAM Journal on Scientific Computing and Springer Computational Geosciences. From 2018 to 2019, he was a postdoctoral researcher at the Max Planck Institute for Dynamics of Complex Technical Systems, and since 2020, he has been holding a position as research scientist at Philips GmbH Innovative Technologies, Germany.

Sara Grundel

Sara Grundel was born in Munich Germany in 1981. She received a Diploma in mathematics from ETH Zurich in 2005 and a Ph. D. degree in mathematics from the Courant Institute of Mathematical Sciences at New York University in New York City in 2011. Since then she has been holding a postdoctoral position at the Max Planck Institute in Magdeburg, Germany. In 2015 she became a research team leader focused on the simulation of energy system.

Appendix A Matricization

We illustrate the concept of matricization using an example given by Kolda and Bader [21]. Let H∈R3×4×2 be a tensor whose mode-1 matricization is

(32)H(1)=147101316192225811141720233691215182124.

Then, the mode-2 and mode-3 matricizations are

(33a)H(2)=123131415456161718789192021101112222324,

(33b)H(3)=123⋯101112131415⋯222324.

For more information about matricization and tensors, we refer to [21] and to previous work on model reduction of quadratic-bilinear systems [6], [7].

Appendix B Low-rank approximation

Given P=RRT where P,R∈Rn×n, we denote by R˜=Tτ(R)∈Rn×l a low-rank approximation for which R˜R˜T≈RRT=P, i. e., the purpose is to approximate P. We compute R˜ using the singular value decomposition R=UΣV:

(34)R˜=UΣl.

Here, Σl contains the first l columns of Σ, and l is chosen such that

(35)σi2>τσ12,

for i=2,…,l. The singular values σi are the diagonal entries of Σ, and they are ordered, i. e., σi≥σj for i<j.

In this work, we use the machine precision as the tolerance, i. e., τ=1.1102·10−16, in order to limit the error of the low-rank approximation.

Appendix C Efficient evaluation of the Kronecker products

We evaluate ΔKi,k in (27a) using matricization [21], [6], [7]:

(36a)ΔKi,k=KK(1),

(36b)KK(3)=R˜kTYK(3),

(36c)YK(2)=R˜i−(k+1)TH(2).

Similarly, we evaluate ΔLi,k in (27b) by

(37a)ΔLi,k=KL(2),

(37b)KL(3)=R˜kTYL(3),

(37c)YL(1)=S˜i−(k+1)TH.

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Received: 2020-05-03

Accepted: 2020-10-15

Published Online: 2020-11-27

Published in Print: 2020-11-18

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Keywords for this article

nonlinear model reduction; balanced truncation; dynamical power grid models; quadratic systems